For the parametric representation x=f(t), y=g(t),
dy/dx = (dy/dt)/(dx/dt) （dx/dt≠0）
Physically, (f′(t), g′(t)) is the velocity vector at time t.
Example ① Find the slope of the tangent to x=t², y=t³−t at t=1.
Solution: dx/dt=2t=2, dy/dt=3t²−1=2. dy/dx = 2/2 = 1
Solution: dx/dt=2t=2, dy/dt=3t²−1=2. dy/dx = 2/2 = 1
Key Point
- At points where the denominator dx/dt is 0, the tangent is parallel to the y-axis.
Practice Problems
- Find the slope of the tangent to x=cos t, y=sin t at t=π/4.
Solutions and Explanations
- Solution: −1
dy/dx = cos t/(−sin t), which is −1 at t=π/4.